Newton-like polynomials of links (Q865261)

The author proves the following result: Any root of the one variable Conway polynomial \(\nabla_L(\sqrt{z})\) of a \textsl{special alternating} link \(L\) belongs to the interval \([-4,0]\). Equivalently, the roots of its Alexander polynomial have modulus \(1\). Recall that an oriented (non split) link is \textsl{alternating} if it admits a planar diagram with alternating crossings, and it is \textsl{special} if it admits a digram in which all the \textsl{regions} (i.e. connected components of the complement of the projection of the link) are bounded by edges whose orientations either induce a global orientation of the boundary or alternate along the boundary. The result of the paper proves, for positive links which are also alternating, part (2) of the following conjecture proposed by the author: (1) The Alexander polynomial of an alternating link is \textsl{weakly Newton-like}; (2) The one variable Conway polynomial of a positive link with \(n\) components (times \(z^{(1-n)/2}\)) is \textsl{Newton like}. The conjecture is known to be true for knots up to \(16\) crossings, and its part (1) is a strengthening of a conjecture due to Fox (``trapezoidal conjecture''). Note that a real polynomial of the form \(a_l z^l+a_{l+1}z^{l+1}+\dots+a_nz^n\), with \(a_l,a_n\neq0\), is called \textsl{Newton-like} (respectively \textsl{weakly Newton-like}) if, for all \(l<i<n\) one has \(a_{i-1}a_{i+1}<a_i^2\) (respectively \(a_{i-1}a_{i+1}\leq a_i^2\)).